Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 167 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.14·5-s + 3.32·7-s + 9-s + 2.04·11-s + 5.67·13-s + 1.14·15-s − 4.44·17-s − 4.57·19-s − 3.32·21-s + 8.33·23-s − 3.68·25-s − 27-s − 6.09·29-s − 10.8·31-s − 2.04·33-s − 3.81·35-s + 3.37·37-s − 5.67·39-s + 3.28·41-s − 10.1·43-s − 1.14·45-s − 7.22·47-s + 4.05·49-s + 4.44·51-s + 8.07·53-s − 2.34·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.512·5-s + 1.25·7-s + 0.333·9-s + 0.616·11-s + 1.57·13-s + 0.296·15-s − 1.07·17-s − 1.04·19-s − 0.725·21-s + 1.73·23-s − 0.736·25-s − 0.192·27-s − 1.13·29-s − 1.94·31-s − 0.355·33-s − 0.644·35-s + 0.555·37-s − 0.908·39-s + 0.512·41-s − 1.54·43-s − 0.170·45-s − 1.05·47-s + 0.578·49-s + 0.622·51-s + 1.10·53-s − 0.316·55-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8016} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 8016,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;167\}$, \[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;167\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 + T \)
good5 \( 1 + 1.14T + 5T^{2} \)
7 \( 1 - 3.32T + 7T^{2} \)
11 \( 1 - 2.04T + 11T^{2} \)
13 \( 1 - 5.67T + 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 + 4.57T + 19T^{2} \)
23 \( 1 - 8.33T + 23T^{2} \)
29 \( 1 + 6.09T + 29T^{2} \)
31 \( 1 + 10.8T + 31T^{2} \)
37 \( 1 - 3.37T + 37T^{2} \)
41 \( 1 - 3.28T + 41T^{2} \)
43 \( 1 + 10.1T + 43T^{2} \)
47 \( 1 + 7.22T + 47T^{2} \)
53 \( 1 - 8.07T + 53T^{2} \)
59 \( 1 - 0.491T + 59T^{2} \)
61 \( 1 + 1.93T + 61T^{2} \)
67 \( 1 + 5.76T + 67T^{2} \)
71 \( 1 + 8.42T + 71T^{2} \)
73 \( 1 + 14.6T + 73T^{2} \)
79 \( 1 + 0.640T + 79T^{2} \)
83 \( 1 + 12.9T + 83T^{2} \)
89 \( 1 + 13.8T + 89T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.34626708297727143578128956971, −6.89885041439785179774916785444, −6.01761044967069696847653185953, −5.46847075375512243901701678105, −4.50200288225099855392176351713, −4.13680803895752845516544634855, −3.29907607280752319332932799305, −1.87724672239813832801328460513, −1.37006065342879017111405049345, 0, 1.37006065342879017111405049345, 1.87724672239813832801328460513, 3.29907607280752319332932799305, 4.13680803895752845516544634855, 4.50200288225099855392176351713, 5.46847075375512243901701678105, 6.01761044967069696847653185953, 6.89885041439785179774916785444, 7.34626708297727143578128956971

Graph of the $Z$-function along the critical line